The stock price is assumed to evolve as $S_{t}=S_{0}+\mu t+\sigma B_{t}$, where $S_{0}>0, \mu>0$ and the process $B_{t}$ is Brownian motion.
The saving account is assumed to be $\beta_{t}=e^{r t}$, with interest rate $r$
A call option with strike $K$ and expiration $T$ pays $C_{T}=\left(S_{T}-K\right)^{+}$ at time $T$.
Assume r = 0. Give the EMM.
My attempt
I am a bit lost when it comes to EMM but this is what I have so far:
- Girsanov's theorem:
$B_{t}$ is a B.M under measure P and C is a constant. Then there exists an EMM q such that:
$\hat{B}_{t}=B_{t}+C_{t} \sim Q$ brownian motion.
$d S_{t}=\mu d t+\sigma d B_{t}$
$r=0 \quad c=\frac{\mu-r}{\sigma}=c=\frac{\mu}{\sigma}$
I am not sure how to continue and give the EMM explicitly.
Edit: After some researching, I have found that the EMM does exist for Bachelier model and is unique by Girsanov's theorem. But I am still a bit lost on how to find it.